Suppose $z_1,z_2,\cdots,z_n$ be $n$ complex numbers and $$ r=\max | z_i-z_j|,\;i,j=1,2,\dots,n\;i \neq j.$$Further let $$z=\frac{z_1+z_2+\cdots+ z_n}{n}.$$ Is it true that for all $k=1,2,..,n$,$$ |z-z_k| \leq r?$$ If so how can we prove it? Any hints or suggestions will be highly appreciated.
Asked
Active
Viewed 59 times
1
-
1Yes, it's true. Try playing about with $|n\cdot z- n\cdot z_k|$ and apply the triangle inequality. – Jaap Scherphuis Jun 17 '22 at 10:03
-
Hint: Substitute your formula for $z$ in $|z-z_k|$ and apply the triangle inequality. – Marcos Jun 17 '22 at 10:03
-
Got it!thank you – AgnostMystic Jun 17 '22 at 10:22